by Johan L Dupont (University of Aarhus, Denmark)

These lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume “scissors-congruent”, i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristic classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time.

• Introduction and History
• Scissors Congruence Group and Homology
• Homology of Flag Complexes
• Translational Scissors Congruences
• Euclidean Scissors Congruences
• Sydler's Theorem and Non-Commutative Differential Forms
• Spherical Scissors Congruences
• Hyperbolic Scissors Congruence
• Homology of Lie Groups Made Discrete
• Invariants
• Simplices in Spherical and Hyperbolic 3-Space
• Rigidity of Cheeger-Chern-Simons Invariants
• Projective Configurations and Homology of the Projective Linear Group
• Homology of Indecomposable Configurations
• The Case of PGl(3,F)